```Date: Mar 13, 2013 3:09 PM
Author: Zaljohar@gmail.com
Subject: Re: Reducing Incomparability in Cardinal comparisons

On Mar 12, 10:46 pm, Zuhair <zaljo...@gmail.com> wrote:> On Mar 12, 2:49 pm, Zuhair <zaljo...@gmail.com> wrote:>>>>>>>>>> > On Mar 11, 11:17 pm, Zuhair <zaljo...@gmail.com> wrote:>> > > Let x-inj->y stands for there exist an injection from x to y and there> > > do not exist a bijection between them; while x<-bij-> means there> > > exist a bijection between x and y.>> > > Define: |x|=|y| iff  x<-bij->y>> > > Define: |x| < |y| iff  x-inj->y Or Rank(|x|) -inj-> Rank(|y|)>> > > Define: |x| > |y| iff |y| < |x|>> > > Define: |x| incomparable to |y| iff ~|x|=|y| & ~|x|<|y| & ~|x|>|y|>> > > where |x| is defined after Scott's.>> > > Now those are definitions of what I call "complex size comparisons",> > > they are MORE discriminatory than the ordinary notions of cardinal> > > comparisons. Actually it is provable in ZF that for each set x there> > > exist a *set* of all cardinals that are INCOMPARABLE to |x|. This of> > > course reduces incomparability between cardinals from being of a> > > proper class size in some models of ZF to only set sized classes in> > > ALL models of ZF.>> > > However the relation is not that natural at all.>> > > Zuhair>> > One can also use this relation to define cardinals in ZF.>> > |x|={y| for all z in TC({y}). z <* x}>> > Of course <* can be defined as:>> > x <* y iff [x -inj->y Or> > Exist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) in> > rank(y*)].>> > Zuhair>> All the above I'm sure of, but the following I'm not really sure of:>> Perhaps we can vanquish incomparability altogether>> If we prove that for all x there exist H(x) defined as the set of all> sets hereditarily not strictly supernumerous to x. Where strict> subnumerousity is the converse of relation <* defined above.>> Then perhpas we can define a new Equinumerousity relation as:>> x Equinumerous to y iff H(x) bijective to H(y)>> Also a new subnumerousity relation may be defined as:>> x Subnumerous* to y iff H(x) injective to H(y)Better would bex Subnumerous* to y iff H(x) <* H(y)however still this won't concur incomparability completelyHowever if we define recursively H_n(x) then we can definethe above relations after those. However still incomparabilitywould persist, although the above is still a strong approachagainst incomparability.>> This might resolve all incomparability issues (I very highly doubt> it).>> Then the Cardinality of a set would be defined as the set of all sets> Equinumerous to it of the least possible rank.>> A Scott like definition, yet not Scott's.>> Zuhair
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